Or as I like to spell it, Oiler.



For when the bus gets squeaky and hard to steer ...

But Euler's identity is awesome: the five most important numbers related by the three most important operations.

Last edited Fri Jul 26, 2013, 10:16 AM - Edit history (1)

where tau ( т ) = 2?...

[center]e^iт = 1[/center]

the other form also includes the additive identity. Much more beautiful.

I've never really understood the advantages of tau over pi. Yes, 90 degrees is tau/4 versus pi/2, but when you start to evaluate infinite series you have tau/4 instead of pi/2. Uglier, IMO.

I've even heard the argument (not from you, pokerfan) that using the word which sounds like "pie" can be confusing because we think of a pie as being a circle, not a semicircle. Trouble is, the Greeks pronounce it "pee". Nice.

If you must have zero and addition....

[center]e^iт = 1 + 0

[/center]

The advantage is that it's much more intuitive. Instead of one pi radians equaling just half a turn around the unit circle, which takes you to (-1,0), one tau equals one turn. Simple and clear.

It's idiotic to use pi (which is based on the diameter) for the circle constant while measuring angles with radians (which is based on the radius). It's mixing metaphors.

Alternatively, we could keep pi but change the radian to something based on the diameter. That would also be consistent.

1 + 0????? Give me a break.

I very much doubt that tau will ever take off.

You didn't address my comment about infinite series adding up to quantities such as pi/2, pi^2/8. The tau versions are no more intuitive but are uglier.

from e^i*pi = -1?

All they did to make "the most beautiful equation" in the world is add 1 to both sides. Give me a break. Math should illuminate, not obfuscate and mixing diameters and radii in the same formula is very, very ugly.

What specific infinite series are you referring to? Gregory & Gottfried Leibniz: Pi = 4 (1 - 1/3 +1/5 - 1/7 + 1/9 - ...) Simply multiply by 2 to express it in tau. It's still a beautiful series either way.

It adds nothing to knowledge. It ain't going nowhere.

"What specific infinite series are you referring to? Gregory & Gottfried Leibniz: Pi = 4 (1 - 1/3 +1/5 - 1/7 + 1/9 - ...) Simply multiply by 2 to express it in tau. It's still a beautiful series either way."

So...Pi = 4 (1 - 1/3 +1/5 - 1/7 + 1/9 - ...)
But...tau = 8 (1 - 1/3 +1/5 - 1/7 + 1/9 - ...)

So where exactly do we see the beauty of tau?

And there are lots of other infinite series...lots of...

Want to see the beauty of tau, watch the video.

The beauty is that 3/4 of a circle is 3/4 tau, not 3/2 pi and so on. Face it, as an EE, I work with it on an almost daily basis and it's a pain in the but. And trigonometry would be so much easier to learn if we actually had consistent units.

... a circular definition.

A damned pretty one, however.

Euler's identity is what it is because that's the way the language built.

Gödel's incompleteness theorems are pretty brutal in everything from math, to computer programming, to the theories of evolution.

To put it poetically, most equations branch out like a seed planted and touch the entire universe.

I once took a class from an established professor who was a few weeks late returning from a very remote place. His substitute was a brand new PhD who approached the science of ecology like many economists study markets. He was overly attracted to pretty equations which looked like something out of a classical physics text. He'd write quiz questions that required us to use calculus.

When the professor returned from his travels and took back his class his first words to us were something like, "Well, this is crap..." It's possible he was even more direct than that, since he was the sort of fellow who'd dug his own holes to shit in, and picked botfly larvae and other parasites out of himself and others. I felt sorry for the poor guy who'd been teaching the class. He probably ended up at a Wall Street bank, or teaching at a community college or high school somewhere in rural Kansas.

Quite suddenly the math we used switched from calculus to statistics, and from laboratory fruit fly experiments ( which tend to be worthless for their selection bias ) to hands-on dirty field work.

Humans are always drawn to the pretty equations, the Musica universalis, but most things in nature are not singing that song.

As John Muir said, "When we try to pick out anything by itself, we find it hitched to everything else in the Universe." ( sierraclub.org )

I'm always suspicious of the answers that are too neat.

I mean, I agree that the Euler identity doesn't really tell us anything about the universe. And that some equations are more fruitful than others in terms of their utility.

But as far as the mathematical validity of any equation goes, that's wholly determined by the definitions of any operators, variables and constants present. That's the sense in which I took you to be calling the identity "tautological," and I think the same argument may be made for any equation.

I totally agree that, on its own, the e^(i pi)-1=0 isn't something you're really going to use to explore the world around us. I'd also hesitate to dismiss calculus entirely, but there's no doubt that it's overemphasized at many levels and a sound grounding in statistics is far more useful in coping with reality!

there is something about this thread ... kind of sexy.

exp(ix) = cos(x) + i sin(x);       (1)

exp(x) = 1 + x + x^2/2! + x^3/3! + ... .       (2)

Here x can be any complex number. Thus the functions exp, sin, cos, sinh, cosh, and other trigonometric and hyperbolic functions can be extended from the real line (or most of it) to the complex plane (or most of it).

Formula 2 can be generalized as follows:

f(x) = f(a) + (x-a) f'(a) + (x-a)^2 f&quot a)/2! + ... .       (3)

provided that f is analytic in a neighborhood of a. This is Taylor's Theorem.

Didn't know the trig functions could be extended to the complex plane neat.

At one point I was going to do a math and or physics degree. I went about as far as 3rd level calculus at university with a number of other courses. I've always been something of a math nerd, and it's always been one of my best subjects at school. Depression and Anxiety issues scratched those plans some years ago and I'm on a different path now but I still love math.

I remember working with Taylor series, quite cool!